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lines B and C are perpendicular. B1","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Identifies Line B and Line C.","code":"B"}],"finalAnswers":["Line B and Line C"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"compare_gradients","label":"Compare gradients","steps":[{"type":"text","order":0,"title":"Compare every gradient","content":"For a line written as $y=mx+c$, the coefficient $m$ is its **gradient** (steepness).\n\nRearrange or read each equation to obtain:\n\n$$m_A=-3,\n\\quad m_B=\\frac12,\n\\quad m_C=-2,\n\\quad m_D=4,\n\\quad m_E=1.$$\n\nPerpendicular lines have gradients whose product is $-1$. Only $\\frac12$ and $-2$ do this, so test Lines B and C.","highlights":[{"text":"Two of these lines are perpendicular.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"m_B=\\htmlData{anchor=mb}{\\color{@sky}{\\frac{1}{2}}}"},{"latex":"m_C=\\htmlData{anchor=mc}{\\color{@orange}{-2}}"},{"latex":"m_B\\times m_C=\\frac{1}{2}\\times(-2)=\\htmlData{anchor=product}{\\color{@green}{-1}}"},{"text":"Therefore, Lines B and C are perpendicular. [B1]"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"product"}]},"order":3,"title":"Test the gradients","description":"Multiply the two gradients. A result of -1 proves the lines are perpendicular. This earns B1."}]}],"generatedAt":"2026-07-21T04:06:51.990Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3e371dc2-7407-45d1-80db-f83b8384f7ae","revisionId":"9bb2f586-fe65-4584-8fec-57436551d0fd","questionNumber":"12","questionType":"LA","specification":"Line A passes through the points $(2, 1)$ and $(5, 10)$ ","difficulty":3,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"8a8baee9-2c62-426c-bfb8-a2d08888af2c","label":"0","content":"Find the equation of the line parallel to A that passes through $(2,5)$","markscheme":"- Calculates gradient of A: $m = \\frac{10 - 1}{5 - 2} = \\frac{9}{3} = 3$ M1\n- Substitutes $(2, 5)$ into $y = 3x + c$: $5 = 3(2) + c \\implies c = -1$ M1\n- Final equation $y = 3x - 1$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Calculates the gradient of Line A as 3.","code":"M"},{"id":"p2","marks":1,"description":"Substitutes the point $(2, 5)$ into $y = 3x + c$ to find $c$.","code":"M"},{"id":"p3","marks":1,"description":"Correct final equation $y = 3x - 1$.","code":"A"}],"finalAnswers":["y=3x-1"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"gradient_then_intercept","label":"Gradient then intercept","steps":[{"type":"text","order":0,"title":"Use parallel gradients","content":"Parallel lines have the same **gradient** (the change in $y$ for each increase of $1$ in $x$). So first find the gradient of line $A$, then use that same gradient for the required line.","highlights":[{"text":"parallel to A","location":"specification"},{"text":"passes through $(2,5)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"m=\\frac{10-1}{5-2}"},{"latex":"m=\\frac{9}{3}"},{"latex":"m=3"}]},"order":1,"title":"Find A's gradient","description":"Use the two given points: subtract the y-coordinates, then divide by the difference in the x-coordinates. This earns the first M1 mark."},{"type":"text","order":2,"title":"Write the line form","content":"A straight-line equation has the form $y=mx+c$, where $c$ is the $y$-intercept. Since the new line is parallel to $A$, substitute $m=3$ to get $y=3x+c$.","highlights":[{"text":"equation of the line parallel to A","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"5=3(2)+c"},{"latex":"5=6+c"},{"latex":"c=-1"}]},"order":3,"title":"Find the intercept","description":"The point (2, 5) lies on the required line, so use x = 2 and y = 5. Substituting this point earns the second M1 mark."},{"type":"text","order":4,"title":"State the equation","content":"Put $c=-1$ into $y=3x+c$:\n\n$$y=3x-1$$\n\nThis is the final answer and earns the A1 mark. Quick check: when $x=2$, the equation gives $y=5$, so it passes through the required point.","highlights":[{"text":"Find the equation","location":"specification"}]}]}],"generatedAt":"2026-07-21T04:03:49.899Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":67},"walkthroughVideo":null},{"id":"279118c0-d7d5-42b6-8a6a-0acb5f408d8f","revisionId":"0cd89d00-1067-47d6-b62f-8189a35b4eec","questionNumber":"13","questionType":"LA","specification":"Line A passes through the points $(1, 5)$ and $(5, 7)$ ","difficulty":3,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"4f873e80-b816-4049-b58a-e5b5080f5c20","label":"0","content":"Find the equation of the line perpendicular to A that passes through $(-1,7)$","markscheme":"- Calculates gradient of A: $m = \\frac{7 - 5}{5 - 1} = \\frac{2}{4} = \\frac{1}{2}$ M1\n- Perpendicular gradient $m = -2$ and substitutes $(-1, 7)$: $7 = -2(-1) + c \\implies c = 5$ M1\n- Final equation $y = -2x + 5$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds the perpendicular gradient $m = -2$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final equation $y = -2x + 5$.","code":"A"}],"finalAnswers":["y=-2x+5"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__cie","summaries":[{"id":"acbe8ebb-46d9-478e-8f4d-932b2885f04c","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"00df6036-9aab-4beb-a1f9-478482a7e43b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"88754053-ce58-485c-a539-99de2a7000a2","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"99e9a58e-58ce-4c5b-bcf8-e1fc35241405","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"5aa9fb76-4e85-46d4-9eb8-828c02ccc7a6","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9b5feb40-0aa0-44e4-8ce6-453faebd020a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e0cb84a5-5545-4566-9b6a-7e816ee630e0","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d258283c-865e-47c0-89da-0e5a1bcf9a81","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"91efa1cb-1437-4764-928f-184d64d102eb","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6277a69d-6c9b-4e4d-a024-c4c43440a622","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"e3704fe4-d3b3-4d05-a5be-25f12b806bf9","difficultyBand":"medium","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"3e371dc2-7407-45d1-80db-f83b8384f7ae","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"279118c0-d7d5-42b6-8a6a-0acb5f408d8f","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c5bf0b85-9272-4d46-9a99-b4887b1793ec","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4bb0fdaa-3145-4c69-b31a-7058b11ddbd8","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a8a661f3-569f-4c77-a493-be9716343848","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c22eabf2-eada-4b44-bdb5-ceda01fdd864","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f5aac4ae-4ceb-4ac8-9cdc-ca59354e843c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2db8e411-8c21-448e-ae58-f85a6e6f3962","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Parallel and Perpendicular Lines","topicId":"697d56f9-6c2e-4572-b3fd-732756250220","topicSidebar":"$L3c","topicSlug":"grade-6-6-12-parallel-and-perpendicular-lines","topicTitle":"Parallel and Perpendicular Lines"}]}],"$L3d","$L3e"]}]

Parallel and Perpendicular Lines

Question 3

Parallel and Perpendicular Lines Questions

Parallel and Perpendicular Lines

Question 3

Parallel and Perpendicular Lines Questions